String Theory Inspires a Brilliant, Baffling New Math Proof
Kristina Armitage/ Quanta Magazine
克里斯蒂娜·阿米蒂奇/广达杂志
I n August, a team of mathematicians posted a paper claiming to solve a major problem in algebraic geometry — using entirely alien techniques.
八月，一组数学家发表了一篇论文，声称使用完全外星的技术解决了代数几何中的一个主要问题。
It instantly captivated the field, stoking excitement in some mathematicians and skepticism in others.
它立即吸引了整个领域，激起了一些数学家的兴奋和另一些数学家的怀疑。
The result deals with polynomial equations, which combine variables raised to powers (like y = x or x 2 − 3 xy = z 2 ).
结果涉及多项式方程，该方程组合了变量的幂（例如 y = x 或 x 2 − 3 xy = z 2 ）。
These equations are some of the simplest and most ubiquitous in mathematics, and today, they’re fundamental to lots of different areas of study.
这些方程是数学中最简单、最普遍的方程，如今，它们是许多不同研究领域的基础。
As a result, mathematicians want to study their solutions, which can be represented as geometric shapes like curves, surfaces and higher-dimensional objects called manifolds .
因此，数学家希望研究他们的解决方案，这些解决方案可以表示为几何形状，如曲线、曲面和称为流形的高维对象。
There are infinitely many types of polynomial equations that mathematicians want to tame.
数学家想要驯服无数种类型的多项式方程。
But they all fall into one of two basic categories — equations whose solutions can be computed by following a simple recipe, and equations whose solutions have a richer, more complicated structure.
但它们都属于两个基本类别之一：解可以通过遵循简单公式计算的方程，以及解具有更丰富、更复杂结构的方程。
The second category is where the mathematical juice is: It’s where mathematicians want to focus their attention to make major advances.
第二类是数学精华所在：数学家希望集中注意力以取得重大进展。
But after sorting just a few types of polynomials into the “easy” and “hard” piles, mathematicians got stuck.
但在将几种类型的多项式分为“简单”和“困难”两类后，数学家们陷入了困境。
For the past half-century, even relatively simple-looking polynomials have resisted classification.
在过去的半个世纪里，即使是看起来相对简单的多项式也无法进行分类。
Then this summer, the new proof appeared.
今年夏天，新的证据出现了。
It claimed to end the stalemate, offering up a tantalizing vision for how to classify lots of other types of polynomials that have until now seemed completely out of reach.
它声称结束了僵局，为如何对许多其他类型的多项式进行分类提供了一个诱人的愿景，而到目前为止，这些多项式似乎完全遥不可及。
The problem is that no one in the world of algebraic geometry understands it.
问题是代数几何领域没有人理解它。
At least, not yet.
至少现在还没有。
The proof relies on ideas imported from the world of string theory.
该证明依赖于从弦理论世界引入的思想。
Its techniques are wholly unfamiliar to the mathematicians who have dedicated their careers to classifying polynomials.
对于那些致力于多项式分类的数学家来说，它的技术是完全陌生的。
Some researchers trust the reputation of one of the paper’s authors, a Fields medalist named Maxim Kontsevich .
一些研究人员相信该论文的作者之一、菲尔兹奖得主马克西姆·康采维奇（Maxim Kontsevich）的声誉。
But Kontsevich also has a penchant for making audacious claims, giving others pause.
但康采维奇也喜欢提出大胆的主张，让其他人犹豫不决。
Reading groups have sprung up in math departments across the world to decipher the groundbreaking result and relieve the tension.
世界各地的数学系纷纷涌现阅读小组，以解读这一突破性的结果并缓解紧张气氛。
This review may take years.
这项审查可能需要数年时间。
But it’s also revived hope for an area of study that had stalled.
但这也让一个陷入停滞的研究领域重燃了希望。
And it marks an early victory for a broader mathematical program that Kontsevich has championed for decades — one that he hopes will build bridges between algebra, geometry and physics.
这标志着康采维奇几十年来一直倡导的更广泛的数学计划的早期胜利——他希望这一计划能够在代数、几何和物理学之间架起桥梁。
“The general perception,” said Paolo Stellari , a mathematician at the University of Milan who was not involved in the work, “is that we might be looking at a piece of the mathematics of the future.”
米兰大学数学家保罗·斯特拉里（Paolo Stellari）没有参与这项工作，他说：“普遍的看法是，我们可能正在研究未来数学的一部分。”
The effort to classify all polynomials deals with the oldest kind of math: solving equations.
对所有多项式进行分类的工作涉及最古老的数学类型：求解方程。
To solve the simple polynomial y = 2 x , for instance, you just need to find values of x and y that satisfy the equation.
例如，要求解简单多项式 y = 2 x ，您只需找到满足方程的 x 和 y 值。
There are infinitely many solutions to this equation, such as x = 1, y = 2.
该方程有无穷多个解，例如 x = 1，y = 2。
When you graph all the solutions in the coordinate plane, you get a line.
当您在坐标平面上绘制所有解决方案时，您会得到一条线。
Other polynomials are harder to solve directly, and their solutions cut out more complicated, higher-dimensional shapes in space.
其他多项式更难直接求解，它们的解在空间中切出更复杂、更高维的形状。
But for some of these equations, it turns out, there’s a really simple way to find every possible solution.
但事实证明，对于其中一些方程，有一种非常简单的方法可以找到所有可能的解。
Instead of separately plugging different numbers into each variable, you can get all the solutions at once by rewriting the variables in terms of a new variable, t .
您无需将不同的数字分别插入每个变量，而是可以通过用新变量 t 重写变量来一次获得所有解决方案。
Consider the polynomial x 2 + y 2 = 1, which defines a circle.
考虑多项式 x 2 + y 2 = 1，它定义了一个圆。
Now set x equal to 2 t /(1 + t 2 ), and y equal to (1 − t 2 )/(1 + t 2 ).
现在设置 x 等于 2 t /(1 + t 2 )，y 等于 (1 − t 2 )/(1 + t 2 )。
When you plug these new formulas back into your original equation, you get 1 = 1, a statement that’s always true, no matter what t is.
当你将这些新公式代入原来的方程时，你会得到 1 = 1，无论 t 是什么，这个陈述总是正确的。
This means that by choosing any real-number value for t , you’ll instantly get a solution to the original polynomial.
这意味着通过为 t 选择任何实数值，您将立即获得原始多项式的解。
For instance, when you set t equal to 1, you get x = 2(1)/(1 + (1) 2 ) = 1, and y = 0.
例如，当您将 t 设置为 1 时，您将得到 x = 2(1)/(1 + (1) 2 ) = 1，并且 y = 0。
And indeed, x = 1, y = 0 is a solution to the original equation: (1) 2 + (0) 2 = 1.
事实上，x = 1, y = 0 是原方程的解：(1) 2 + (0) 2 = 1。
This straightforward way of framing all your solutions is called a rational parameterization.
这种构建所有解决方案的简单方法称为理性参数化。
It’s equivalent to mapping every point on the graph of your original polynomial — in this case, a circle — to a unique point on a straight line.
它相当于将原始多项式图上的每个点（在本例中为圆）映射到直线上的唯一点。
Choose a point on the circle (blue).
在圆上选择一个点（蓝色）。
You want to map it to a unique point on the straight yellow line.
您想将其映射到黄色直线上的唯一点。
To do this, draw a dashed line between the green point at the top of the circle and your chosen blue point.
为此，请在圆圈顶部的绿点和您选择的蓝点之间绘制一条虚线。
Then map the blue point to whichever yellow point this dashed line passes through.
然后将蓝色点映射到该虚线穿过的黄色点。
You can do this for any given point on the circle.
您可以对圆上的任何给定点执行此操作。
(The green point at the top of the circle gets mapped to a special yellow point at infinity.)
（圆圈顶部的绿点被映射到无穷远处的特殊黄点。）
Any degree-1 polynomial equation — that is, any polynomial whose terms are raised to a power of at most 1 — can be parameterized like this.
任何 1 次多项式方程——即任何项最多为 1 次方的多项式——都可以这样参数化。
It doesn’t matter how many variables the equation has: It might have two variables, or 200.
方程有多少个变量并不重要：它可能有两个变量，或者 200 个。
Once you go beyond two variables, the solutions to your polynomial equation will form complicated higher-dimensional shapes.
一旦超出两个变量，多项式方程的解将形成复杂的高维形状。
But because the polynomial can still be parameterized, there’s a way to map every point in your high-dimensional shape to points on a particularly simple space in the same number of dimensions (like the line).
但由于多项式仍然可以参数化，因此有一种方法可以将高维形状中的每个点映射到相同维数（如直线）的特别简单空间上的点。
This, in turn, gives you a straightforward way to compute all the polynomial’s solutions.
反过来，这为您提供了一种计算所有多项式解的简单方法。
Similarly, any degree-2 polynomial (whose terms are raised to a power of at most 2) has a rational parameterization.
类似地，任何 2 次多项式（其项最多为 2 次方）都具有有理参数化。
But if an equation’s degree is 3 or more, it can’t always be parameterized.
但如果方程的次数为 3 或更高，则不能总是参数化。
It depends on how many variables the equation has.
这取决于方程有多少个变量。
Take a typical kind of degree-3 polynomial: elliptic curves, like y 2 = x 3 + 1, which have only two variables.
采用一种典型的 3 次多项式：椭圆曲线，如 y 2 = x 3 + 1，它只有两个变量。
“Elliptic curves are glorious, they’re wonderful, but you can’t possibly parameterize them,” said Brendan Hassett of Brown University.
布朗大学的布伦丹·哈塞特说：“椭圆曲线很光荣，它们很棒，但你不可能对它们进行参数化。”
There’s no simple formula for x and y that gives you all of an elliptic curve’s solutions, so there’s no way to map the curve to a straight line.
没有简单的 x 和 y 公式可以为您提供椭圆曲线的所有解，因此无法将曲线映射为直线。
“If you could, they would not be so much fun,” Hassett said.
“如果可以的话，它们就不会那么有趣了，”哈塞特说。
Unlike in the previous examples, your dashed line sometimes maps two different points on the elliptic curve (blue) to the same point on the yellow line below.
与前面的示例不同，虚线有时会将椭圆曲线（蓝色）上的两个不同点映射到下面黄线上的同一点。
You can’t find a map that avoids this, meaning that the elliptic curve has a more complicated set of solutions than the circle or sphere.
你找不到可以避免这种情况的地图，这意味着椭圆曲线有一组比圆或球体更复杂的解。
Instead, the solutions to an elliptic curve have a far richer structure — one that’s played a vital role in number theory for centuries, and that cryptographers have taken advantage of to encode secret messages.
相反，椭圆曲线的解具有更丰富的结构——几个世纪以来，它在数论中发挥着至关重要的作用，密码学家已经利用它来编码秘密消息。
What about degree-3 equations with more variables, then?
那么具有更多变量的 3 次方程呢？
Are they parameterizable, or is the structure of their solutions more fun, the way it is for elliptic curves?
它们是可参数化的，还是它们的解决方案的结构更有趣，就像椭圆曲线一样？
In 1866, the German mathematician Alfred Clebsch showed that degree-3 equations with three variables — whose solutions form two-dimensional surfaces — are usually parameterizable.
1866 年，德国数学家 Alfred Clebsch 证明，具有三个变量的 3 次方程（其解形成二维表面）通常是可参数化的。
More than a century later, Herbert Clemens and Phillip Griffiths published a monumental proof in which they showed that the opposite is true for most degree-3 equations with four variables.
一个多世纪后，赫伯特·克莱门斯 (Herbert Clemens) 和菲利普·格里菲斯 (Phillip Griffiths) 发表了一项具有里程碑意义的证明，其中他们表明，对于大多数具有四个变量的 3 次方程来说，情况恰恰相反。
These equations, which form three-dimensional manifolds called three-folds, are not parameterizable : Their solutions can’t be mapped to a simple 3D space.
这些方程形成称为三重的三维流形，不可参数化：它们的解无法映射到简单的 3D 空间。
Many mathematicians suspected that the next polynomial to be classified — degree-3 equations with five variables (forming four-dimensional manifolds known as four-folds) — wouldn’t usually be parameterizable either.
许多数学家怀疑下一个要分类的多项式——具有五个变量的三阶方程（形成称为四重的四维流形）——通常也不能参数化。
In fact, they figured that polynomials should never be parameterizable past a certain point.
事实上，他们认为多项式永远不应该在超过某个点之后进行参数化。
But Clemens and Griffiths’ techniques didn’t work for four-folds.
但克莱门斯和格里菲斯的技术对四倍不起作用。
And so for decades, the classification effort lay dormant.
因此几十年来，分类工作一直处于休眠状态。
Mathematicians were surprised when, at a conference in Moscow in the summer of 2019, Maxim Kontsevich got up to speak about classifying four-folds.
2019 年夏天，在莫斯科举行的一次会议上，马克西姆·康采维奇 (Maxim Kontsevich) 站起来谈论四重分类，令数学家们感到惊讶。
For one thing, Kontsevich is known for taking a high-level approach to mathematics, preferring to pose ambitious conjectures and sketch out broad programs, often leaving the subtler details and formal proof-writing to others.
一方面，康采维奇以采用高水平的数学方法而闻名，更喜欢提出雄心勃勃的猜想并勾勒出广泛的计划，通常将更微妙的细节和正式的证明写作留给其他人。
He described his role on a recent project as something between a prophet and a daydreamer.
他将自己在最近的一个项目中扮演的角色描述为介于先知和白日梦者之间的角色。
Maxim Kontsevich often prefers thinking about broad mathematical visions rather than individual problems.
马克西姆·康采维奇通常更喜欢思考广泛的数学视野，而不是单个问题。
For the past three decades, he’s been focused on developing a program called homological mirror symmetry, which has its roots in string theory.
在过去的三十年里，他一直致力于开发一种名为同调镜像对称的程序，该程序植根于弦理论。
In the 1980s, string theorists wanted to count the number of curves on high-dimensional manifolds to answer questions about how the building blocks of the universe might behave.
在 20 世纪 80 年代，弦理论学家想要计算高维流形上的曲线数量，以回答有关宇宙构建块可能如何表现的问题。
To count the curves on a given manifold, they considered its “mirror image” — another manifold that, though very different from the original, had related properties.
为了计算给定流形上的曲线，他们考虑了它的“镜像”——另一个流形，虽然与原始流形非常不同，但具有相关的属性。
In particular, they found that an algebraic object associated to the mirror image, called a Hodge structure, could reveal the number of curves on the original manifold.
特别是，他们发现与镜像相关的代数对象（称为霍奇结构）可以揭示原始流形上的曲线数量。
The reverse was also true: If you count the curves on the mirror image, you’ll get information about the original manifold’s Hodge structure.
反之亦然：如果计算镜像上的曲线，您将获得有关原始流形霍奇结构的信息。
In 1994, Kontsevich sketched out a program to explain the underlying reason for this correspondence.
1994 年，Kontsevich 草拟了一个程序来解释这种通信的根本原因。
His program also predicted that the correspondence extended to all kinds of manifolds beyond those relevant to string theory.
他的程序还预测，这种对应关系会扩展到与弦理论相关的各种流形。
We might be looking at a piece of the mathematics of the future.
我们可能正在研究未来的数学。
Paolo Stellari, University of Milan
保罗·斯特拉里，米兰大学
For now, no one knows how to prove Kontsevich’s mirror symmetry program.
目前，没有人知道如何证明康采维奇的镜像对称纲领。
“It will be next-century mathematics,” he said.
“这将是下世纪的数学，”他说。
But over the years, he’s made partial progress toward a proof — while also exploring the program’s potential consequences.
但多年来，他在证明方面取得了部分进展，同时也探索了该计划的潜在后果。
In 2002, one of Kontsevich’s friends, Ludmil Katzarkov of the University of Miami, hypothesized one such consequence: that the program might be relevant to the classification of polynomial equations.
2002 年，Kontsevich 的一位朋友、迈阿密大学的 Ludmil Katzarkov 假设了这样一个结果：该程序可能与多项式方程的分类相关。
Katzarkov was familiar with Clemens and Griffiths’ 1972 proof that three-folds aren’t parameterizable.
Katzarkov 熟悉 Clemens 和 Griffiths 1972 年证明三重不可参数化的证明。
In that work, the pair looked at a given three-fold’s Hodge structure directly.
在这项工作中，两人直接研究了给定的三重霍奇结构。
They then used it to show that the three-fold couldn’t be mapped to a simple 3D space.
然后他们用它来证明三重不能映射到简单的 3D 空间。
But the Hodge structures associated with four-folds were too complicated to analyze using the same tools.
但与四重相关的霍奇结构太复杂，无法使用相同的工具进行分析。
Katzarkov’s idea was to access the four-fold’s Hodge structure indirectly — by counting how many curves of a particular type lived on its mirror image.
卡察科夫的想法是间接访问四重的霍奇结构——通过计算有多少特定类型的曲线存在于其镜像上。
Typically, mathematicians studying the Hodge structures of four-folds don’t think about curve counts like these: They only come up in seemingly unrelated areas of math, like string theory.
通常，研究四重霍奇结构的数学家不会考虑这样的曲线计数：它们只出现在看似不相关的数学领域，例如弦理论。
But if the mirror symmetry program is true, then the number of curves on the mirror image should illuminate features of the original four-fold’s Hodge structure.
但如果镜像对称程序成立，那么镜像上的曲线数量应该能够阐明原始四重霍奇结构的特征。
Ludmil Katzarkov has argued for decades that mirror symmetry, an ambitious mathematical program inspired by physics, holds the key to solving a major open problem in algebraic geometry.
几十年来，卢德米尔·卡察科夫（Ludmil Katzarkov）一直认为，镜像对称是一个受物理学启发的雄心勃勃的数学程序，它是解决代数几何中一个重大开放问题的关键。
In particular, Katzarkov wanted to break the mirror image’s curve count into pieces, then use the mirror symmetry program to show that there was a corresponding way to break up the four-fold’s Hodge structure.
特别是，卡察科夫想将镜像的曲线计数分解成碎片，然后使用镜像对称程序来证明存在相应的方法来分解四重的霍奇结构。
He could then work with these pieces of the Hodge structure, rather than the whole thing, to show that four-folds can’t be parameterized.
然后，他可以使用霍奇结构的这些部分，而不是整个结构，来证明四重不能参数化。
If any one of the pieces couldn’t be mapped to a simple 4D space, he’d have his proof.
如果其中任何一件作品无法映射到简单的 4D 空间，他就会有证据。
But this line of reasoning depended on the assumption that Kontsevich’s mirror symmetry program was true for four-folds.
但这种推理依赖于这样的假设：康采维奇的镜像对称程序对于四重来说是正确的。
“It was clear that it should be true, but I didn’t have the technical ability to see how to do it,” Katzarkov said.
“很明显，这应该是真的，但我没有技术能力知道如何做到这一点，”卡扎科夫说。
He knew someone who did have that ability, though: Kontsevich himself.
不过，他认识一个确实有这种能力的人：康采维奇本人。
But his friend wasn’t interested.
但他的朋友不感兴趣。
For years, Katzarkov tried to convince Kontsevich to apply his research on mirror symmetry to the classification of polynomials — to no avail.
多年来，卡察科夫试图说服康采维奇将他对镜像对称的研究应用于多项式的分类，但没有成功。
Kontsevich wanted to focus on the whole program, not this particular problem.
康采维奇想要关注整个计划，而不是这个特定的问题。
Then in 2018, the pair, along with Tony Pantev of the University of Pennsylvania, worked on another problem that involved breaking Hodge structures and curve counts into pieces.
然后在 2018 年，两人与宾夕法尼亚大学的托尼·潘特夫 (Tony Pantev) 一起研究了另一个问题，涉及将霍奇结构和曲线计数分解成碎片。
It convinced Kontsevich to hear Katzarkov out.
它说服了康采维奇听取卡察科夫的发言。
Katzarkov walked him through his idea again.
卡扎科夫再次向他阐述了他的想法。
Immediately, Kontsevich discovered an alternative path that Katzarkov had long sought but never found: a way to draw inspiration from mirror symmetry without actually relying on it.
康采维奇立即发现了卡扎科夫长期以来一直在寻找但从未找到的替代途径：一种从镜像对称中汲取灵感而不实际依赖它的方法。
“After you’ve spent years thinking about this, you see it happening in seconds,” Katzarkov said.
卡扎科夫说：“当你花了数年时间思考这个问题后，你会发现它在几秒钟内发生了。”
“That’s a spectacular moment.”
“那是一个壮观的时刻。”
Tony Pantev studies the structure of manifolds by holding them in front of a mathematical mirror.
托尼·潘特夫（Tony Pantev）通过将流形放在数学镜子前来研究流形的结构。
Kontsevich argued that it should be possible to use the four-fold’s own curve counts — rather than those of its mirror image — to break up the Hodge structure.
康采维奇认为，应该可以使用四重自身的曲线计数（而不是其镜像的计数）来打破霍奇结构。
They just had to figure out how to relate the two in a way that gave them the pieces they needed.
他们只需要弄清楚如何以某种方式将两者联系起来，为他们提供所需的部分。
Then they’d be able to focus on each piece (or “atom,” as they called it) of the Hodge structure separately.
然后他们就能够分别关注霍奇结构的每个部分（或他们所说的“原子”）。
This was the plan Kontsevich laid out for his audience at the 2019 conference in Moscow.
这是 Kontsevich 在 2019 年莫斯科会议上为观众制定的计划。
To some mathematicians, it sounded as though a rigorous proof was just around the corner.
对于一些数学家来说，严格的证明似乎指日可待。
Mathematicians are a conservative bunch and often wait for absolute certainty to present new ideas.
数学家是一群保守的人，经常等待绝对确定的情况来提出新想法。
But Kontsevich has always been a little bolder.
但康采维奇总是更大胆一些。
“He’s very open with his ideas, and very forward-thinking,” said Daniel Pomerleano , a mathematician at the University of Massachusetts, Boston, who studies mirror symmetry.
“他的想法非常开放，而且非常具有前瞻性，”波士顿马萨诸塞大学研究镜像对称的数学家丹尼尔·波默莱亚诺 (Daniel Pomerleano) 说。
There was a major ingredient they still had no idea how to address, Kontsevich warned: a formula for how each atom would change as mathematicians tried to map the four-fold to new spaces.
康采维奇警告说，有一个主要因素他们仍然不知道如何解决：当数学家试图将四重映射到新空间时，每个原子将如何变化的公式。
Only with such a formula in hand could they prove that some atom would never reach a state corresponding to a properly “simplified” four-fold.
只有掌握了这样的公式，他们才能证明某些原子永远不会达到与适当“简化”的四重相对应的状态。
This would imply that four-folds weren’t parameterizable, and that their solutions were rich and complicated.
这意味着四倍是不可参数化的，并且它们的解决方案丰富而复杂。
“But people somehow got the impression that he said it was done,” Pomerleano said, and they expected a proof soon.
“但人们不知何故得到了他说已经完成的印象，”波梅莱诺说，他们预计很快就会有证据。
When that didn’t come to pass, some mathematicians began to doubt that he had a real solution.
当这没有实现时，一些数学家开始怀疑他是否有真正的解决方案。
In the meantime, Tony Yue Yu , then at the French National Center for Scientific Research, joined the team.
与此同时，时任法国国家科学研究中心的托尼·余跃也加入了该团队。
Yu’s fresh insights and meticulous style of proof, Kontsevich said, turned out to be crucial to the project.
康采维奇说，余的新颖见解和细致的证明风格对该项目至关重要。
They say, ‘This is black magic, what is this machinery?’
他们说，“这是黑魔法，这是什么机器？”
Maxim Kontsevich, Institute for Advanced Scientific Studies (IHES)
Maxim Kontsevich，高级科学研究所 (IHES)
When lockdowns began during the Covid pandemic, Yu visited Kontsevich at France’s nearby Institute for Advanced Scientific Studies (IHES).
当新冠病毒大流行期间开始封锁时，余拜访了附近法国高级科学研究所（IHES）的康采维奇。
They relished the quiet of the deserted institute, spending hours in lecture halls where there were more blackboards, Yu recalled.
于回忆说，他们喜欢在废弃研究所的安静中度过，在黑板更多的报告厅里待了几个小时。
Meeting regularly with Pantev and Katzarkov over Zoom, they quickly completed the first part of their proof, figuring out precisely how to use the number of curves on a given four-fold to break its Hodge structure into atoms.
他们通过 Zoom 定期与 Pantev 和 Katzarkov 会面，很快完成了证明的第一部分，准确地弄清楚如何使用给定四重上的曲线数量将其 Hodge 结构分解为原子。
But they struggled to find a formula to describe how the atoms could then be transformed.
但他们努力寻找一个公式来描述原子如何转变。
What they didn’t know was that a mathematician who had attended Kontsevich’s lecture in Moscow — Hiroshi Iritani of Kyoto University — had also started pursuing such a formula.
他们不知道的是，一位在莫斯科听过康采维奇讲座的数学家——京都大学的入谷浩史——也开始研究这样的公式。
“He was enchanted by my conjecture,” Kontsevich said.
“他对我的猜想着迷，”康采维奇说。
“I didn’t know, but he started to work on it.”
“我不知道，但他开始努力了。”
In July 2023, Iritani proved a formula for how the atoms would change as four-folds were mapped to new spaces.
2023 年 7 月，入谷证明了一个公式，说明当四倍映射到新空间时，原子会如何变化。
It didn’t give quite as much information as Kontsevich and his colleagues needed, but over the next two years, they figured out how to hone it.
它没有提供 Kontsevich 和他的同事所需的那么多信息，但在接下来的两年里，他们找到了如何完善它的方法。
They then used their new formula to show that four-folds would always have at least one atom that couldn’t be transformed to match simple 4D space.
然后，他们使用新公式证明，四重总是至少有一个原子无法转换以匹配简单的 4D 空间。
Four-folds weren’t parameterizable.
四倍是不可参数化的。
Tony Yue Yu’s meticulous attention to detail and fresh insights were essential to resolving an important question about polynomial equations, according to his colleagues.
据他的同事称，托尼·余宇对细节的一丝不苟和新颖的见解对于解决有关多项式方程的重要问题至关重要。
When the team posted their proof in August, many mathematicians were excited.
当团队在八月份发布他们的证明时，许多数学家都感到兴奋。
It was the biggest advance in the classification project in decades, and hinted at a new way to tackle the classification of polynomial equations well beyond four-folds.
这是分类项目几十年来最大的进步，并暗示了一种解决远远超出四重的多项式方程分类的新方法。
But other mathematicians weren’t so sure.
但其他数学家并不那么确定。
Six years had passed since the lecture in Moscow.
距离莫斯科的演讲已经过去了六年。
Had Kontsevich finally made good on his promise, or were there still details to fill in?
康采维奇最终兑现了他的诺言，还是还有细节需要填写？
And how could they assuage their doubts, when the proof’s techniques were so completely foreign — the stuff of string theory, not polynomial classification?
当证明的技术完全是外来的——弦理论的东西，而不是多项式分类时，他们如何才能减轻他们的疑虑呢？
“They say, ‘This is black magic, what is this machinery?’” Kontsevich said.
“他们说，‘这是黑魔法，这是什么机器？’”康采维奇说。
“Suddenly they come with this completely new approach, using tools that were previously widely believed to have nothing to do with this subject,” said Shaoyun Bai of the Massachusetts Institute of Technology.
麻省理工学院的白少云说：“突然间，他们提出了这种全新的方法，使用了以前被广泛认为与该主题无关的工具。”
“The people who know the problem don’t understand the tools.”
“了解问题的人并不了解这些工具。”
Bai is one of several mathematicians now trying to bridge this gap in understanding.
白是目前试图弥合这一理解差距的几位数学家之一。
Over the past few months, he has co-organized a “reading seminar” made up of graduate students, postdoctoral researchers and professors who hope to make sense of the new paper.
在过去的几个月里，他与他人共同组织了一场由研究生、博士后研究人员和教授组成的“阅读研讨会”，他们希望能够理解这篇新论文。
Each week, a different mathematician digs into some aspect of the proof and presents it to the rest of the group.
每周，一位不同的数学家都会深入研究证明的某些方面，并将其呈现给小组的其他成员。
But even now, after 11 of these 90-minute sessions, the participants still feel lost when it comes to major details of the proof.
但即使是现在，在 90 分钟的会议中进行了 11 次之后，参与者在谈到证明的主要细节时仍然感到茫然。
“The paper contains brilliant original ideas,” Bai said, which “require substantial time to absorb.”
“这篇论文包含了精彩的原创想法，”白说，“需要大量的时间来吸收。”
Similar reading groups have been congregating in Paris, Beijing, South Korea and elsewhere.
类似的阅读团体已经聚集在巴黎、北京、韩国和其他地方。
“People all over the globe are working on the same paper right now,” Stellari said.
“世界各地的人们现在都在研究同一篇论文，”斯特拉里说。
“That’s a special thing.”
“这是一件特别的事情。”
Hassett likens it to Grigori Perelman’s 2003 proof of the Poincaré conjecture, which also used entirely new techniques to solve a famous problem.
哈塞特将其比作 Grigori Perelman 2003 年对庞加莱猜想的证明，该猜想也使用了全新的技术来解决一个著名的问题。
It was only after other mathematicians reproduced Perelman’s proof using more traditional tools that the community truly accepted it.
直到其他数学家使用更传统的工具重现佩雷尔曼的证明之后，社区才真正接受了它。
“There will be resistance,” Katzarkov said, “but we did the work, and I’m sure it’s correct.” He and Kontsevich also see it as a major win for the mirror symmetry program: While they’re not closer to proving it, the result provides further evidence that it’s true.
“会有阻力，”卡扎科夫说，“但我们做了工作，而且我确信这是正确的。”他和 Kontsevich 也认为这是镜像对称计划的重大胜利：虽然他们还没有更接近证明这一点，但结果提供了进一步的证据证明它是正确的。
“I’m very old, and very tired,” Katzarkov said.
“我已经很老了，也很累，”卡察科夫说。
“But I’m willing to develop this theory as long as I’m alive.”
“但只要我还活着，我就愿意发展这个理论。”
Clarification: December 15, 2025 This article has been updated to provide context for a description of Kontsevich as something between a prophet and a daydreamer.
澄清：2025 年 12 月 15 日 本文已更新，为将康采维奇描述为介于先知和白日梦者之间的人物提供了背景。
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